Proceedings of the International Congress of Mathematicians
August 16-24, 1983, Warszawa
SEEGITJ KLATNEEMAE"
Long Time Behaviour of Solutions to Nonlinear
Wave Equations
Most basic equations of both, physics and geometry have the form of
nonlinear, second order, autonomous systems
G(u,u',u") = 0,
(1)
1 2
n+1
where u = u(x , x ,..., x ), and u', u" denote all the first and second
partial derivatives of u. For simplicity we will assume here that both
u and G are scalars and denote by ua, uab, the partial derivatives dau
and respectively d2ahu; a,b =1,2, ..., n+1. Let uQ(x) be a given solution
of (1). Our equation is said to be elliptic or hyperbolic at u°(x) according to whether the (n+l)x(n+^l) matrix, whose entries are Gab
= --—(u°9 u0', u°"), is nondegenerate and has signature (1,...,1,1) or
(1,..., 1, —1). Nonlinear elliptic equations and systems have received a lot
of attention in the past forty or fifty years and in this period a lot of progress
was made and powerful methods were developed. By comparison, the
field of nonlinear hyperbolic equations is wide open. In what follows I
will try to point out some recent developments concerning long-time
behaviour of smooth solutions to a large class of such equations.
Let us assume that (7(0, 0,0) = 0 and that (1) is hyperbolic around
the trivial solution u° = 0. Typically, the operator obtained by linearizing
(1) around u° H== 0 contains only second derivatives. Without further loss
of generality we may assume it to be the wave operator d\ + . . . + d2n — d)
= — D9 where we have ascribed to xn+1 the role of the time variable t.
The equation (1) takes the form (1')
nu =I?(u9u',u")
(!')
n
with F a smooth function of (u,u',u ), independent of uu, vanishing
together with all its first derivatives at (0,0, 0).
[1209]
1210
Section 11: S. KLainerman
Associate to (1') the pure initial value problem
u(x, 0) = sf(x), ut(x, 0) = sg(x) y
(l'a)
with/, g, ö°°-functions, decaying sufficiently fast at infinity (for simplicity,
sa
y ft 9 eG™(Rn)) and e is a parameter which measures the amplitude
of the data. Given /, g and F we define the life span Î7* = T#(e) as the
supremum over all T > 0 such that a <7°°-solution of (1'), (l'a) exists for
all xeRn, 0 < £ < J F . The fundamental local existence theorem ([3], [4],
[14], [24], [27]), asserts that, if s is sufficiently small, so that the initial
data lies in a neighborhood of hyperbolicity of the zero solution, then
T*(s) > 0. Moreover, a simple analysis of the proof shows that T*(e) > A —
8
where A is some small constant, depending only on a finite number of
derivatives of Fr f, g. This lower bound for 2% is in general sharp if the
number of space dimensions n is equal to one. Indeed let our variables in
(1') be x and t and F = o,(%)^œa. with a a smooth function, <r(0) = 0.
An old result of P. Lax [22], extended to systems of wave equations by
F. John [10], shows that, under the assumption of "genuine nonlinearity",
(/(O) ^ 0, all solutions to the corresponding initial value problem (l'a)
blow up by the time O (—). Eecently, in [20], it was proved that 2* < oo
even if the genuine nonlinearity condition is violated. More precisely,
assume that <r'(0) = ... = or(p)(0) = 0, a(p+1)(0) ^ 0 then the corresponding solutions blow up by the time T = 0(1 /e**4"1). In both situations the
blow-up occurs in the second derivatives of u i.e. uxx becomes infinite
while ut, ux remain bounded. Such blow-ups are typical of shock formations
and are observable phenomena of physical reality. If the original equation,
or system can be written in conservation form i.e., in our case,
F(u9u',un)
n+l
= ]? dafa(u,u'),
one can try to^exteiM the solutions past
a=l
these breakdown points by introducing the concept of weak solutions.
This was successfully accomplished for very general first order systems
of conservation laws, in one space dimension, by the fundamental work
of Oleinik, P. Lax and J. Glimm (see [24] for a bibliography). In this
lecture I will restrict myself, however, to classical, i.e. 0°°-solutions.
Suprisingly, the situation looks better in higher dimensions. In 1976
F. John [9] proved that, under the assumption F = F(u', u"), and n^39
T*(s) can be significantly improved and, in 1980, S. Klainerman [15] was
able to push T*(e) to infinity, and thus obtain global solutions, provided
that n^Q. More generally, see ([17], [21], [38]),
Long Time Behavionr of Solutions to Nonlinear Wave Equations
1211
1. Assitme that F = F(u',u") = 0(\u'\ + \u"\Y+l for
1/
1\
n-1
uf, u" and that—I1H
J<
, then there exists an e0 sufficiently small
THEOKEM
such that for all e < £ 0 , (1'), (l'a) has a unique smooth solution for all
x G Rn, t ^ 0.1
The reason for this improved behaviour of solutions of (1') in higher
dimensions was beautifully illustrated by F . John [9] with the following
quotation from Shakespeare, Henry VI:
"Glory is like a circle in the water,
Which never ceaseth to enlarge itself,
Till by broad spreading it disperse to naught".
Indeed, the higher the dimension the more room for waves to disperse and
thus decay. Accordingly, the key to [9], [15], [17], [21] and [28] is to
use decay estimates for solutions to the classical wave equation, • u = 0
(see [25], [26], [31]), and to combine them with energy estimates for
higher derivatives of solutions to the original, nonlinear equations.
The dimension n = 3, which nature gives preference, is not only the
most important but also the most challenging. In [7] F . John exhibited
an example for which T* < oo. More precisely consider F = ut-uu and
the corresponding equation (1') in three space dimensions. Imposing only
one mild restriction on the data, jg(x)dx > 0, F . John showed that there
are no C2-solutions defined for all x e fi3, t ^ 0. However, for sufficiently
small s, the solutions remain smooth for an extremely long-time before
a breakdown occurs. We have in fact the following very general.
THEOKEM 2 (F. John, S. Klainerman). Assume that F verifies one of
the following hypothesis :
(Hx) F does not depend explicitly on u i.e., F = F(u', u")
(H2) F can be written in conservation form,
4
0=1
(R'2) F(u, u', u") = V 8Ja(u,
«0 +0(\u\ + \u'\ + \u"\)3
for small u, u',u
1
The author was recently able to improve this result [32]. The sharp condition which assures global existence is p > 2j(n — 1).
1212
Section 11: S, Klainerman
Then, there exist three small, positive constcmts eö, A9 depending only on
a finite number of derivatives of F9 /, g, such that for every 0 < s < s09
T*( ß )>exp(A/s).
Previously a weaker, polynomially long time existence result, was
proved by F. John in [12] using an asymptotic expansion in powers of
s, for u (see also [9]). The exponential long-time existence result was first
proved, for spherically symmetric solutions (in the semilinear case
F = F(u'), by F. John [7] and T. Sideris [29], and for F = F(u'9 u") by
S. Klainerman [18]).
The result of Theorem 2 is in general sharp. Indeed, F. John [11]
proved recently that this is the case in the context of his previous example,
F = utuu. There is, however, quite a rich class of nonlinearities F for
which global existence holds. The following can be regarded as a generalization of Theorem 1 in dimension n = 3.
TEGEOKEM
3. Assume that F verifies the following "Null"-condition
P) 2 J^(u9u99u9')JPJ?
= 0(\u\+ \u'\ + \u"\?9
( ü ). 2 ^ 0 „ 6c («,«',«")X"z 6 x c
= O(|«I + I«'I+I«"I)',
a,b,c=l
(iii) j ;
KabUJu,u%u")XaXbX°X*=0(\ul
+ \u'\ + \u"\)3
(ISO
for every sufficiently small u9 u', u" and any fixed null space-time vector
(JP,Z«, J 3 ,X*) i.e., (Z1)" + (Z")» + (Z8)»-(2:*)" = 0 .
Then, if e is sufficiently small, a global smooth solution of (!'), (l'a)
To illustrate the content of Theorem 2 note that either of the following
examples verifies the condition (N)
Example 1 F = uaubc — ubuac9 for any three indices a9 6, c = 1, 2, 3, 4
Example 2 F = da(u2 — u\ — u\--u\)9 for any index a = 1 , 2 , 3 , 4 .
The proof of both Theorems 2 and 3 depends on some recent [19]
weighted L°° and L1 estimates for solutions to the classical, inhomogeneous,
wave equation in dimension n = 3. They were first used, in the spherical
symmetric case, in [18] and then extended to the general case by introducing the angular momentum operators Qx = x3d2—x%d39 Q2 =^x1d3 —
— x3d19 Qs = xzd1 — x1dz. Their key property is that they commute with
the wave operator • and thus can be treated as the usual partial deriva-
Long Time Behaviour of Solutions to Nonlinear Wave Equations
1213
tives d19 d2, da. In particular, this allows us to extend the energy estimates
used in [9], [15], [17], [21] and [28], to any combination of the derivatives
^u 5 2 , d3, Q19 Q2, P3. The operators Q19 Q2, AJ are closely connected to
t h e "radiation operators" Z19 L29 L3 which played a fundamental role
m [12].
A different, and very interesting proof of Theorem 3, based on some
conformai mapping methods, was given by D. Christodoulou [1], (See
also his previous joint work with T. Ohoquet-Bruhat [2].)
Both Theorems 2 and 3 have straightforward extensions to systems,
in particular to those of the type arising in Nonlinear Elasticity and General
Eelativity. There are important problems, like that of stability of the
Minkowski Space as a solution of the Einstein equations in vacuum, for
which we hope that Theorems 2 and 3 could be relèvent. In the scalar
case we believe that the picture provided by these theorems, together
with the nonexistence results of F. John [7], [11] can be completed. In
other words, we conjecture that if one of the hypothesis (HI), (H2), (H2')
holds and (N) fails, then the lower bound on T*(e) given by Theorem 2
is sharp, i.e. singularities must develop by that time, for any choice of /
or g and e small. An important open question is to describe the type of
blow-up which occurs in that case. If JP is quasilinear and verifies H I ,
we expect that, as for n =1, the breakdown occurs when the second
derivatives of u become infinite while the first derivatives remain bounded.
The recent work of F. John [28] points in this direction, but completely
satisfactory results are still missing. Another open question is to derive
results similar to Theorems 2 and 3 for the dimensions n = 2 and n = 4.
We suspect that the corresponding, optimal lower bound for I7* for n = 2
must be 01 — J while for n *= 4 one should be able to prove global existence.2
I n this respect we hope to find decay estimates similar to those of [19]
for n T^ 3. The same type of questions can be asked for equations (1')
where the wave operator Q is replaced by the Klein-Gordon operator,
\J + m2 or the Schrödinger operator —id(+A. General results of the type
of Theorem 1 were derived in [17], [21], [28], and for nonlinearities depending only on u in [30], (see also the reference there). The methods
used to derive Theorems 2 and 3 might be used to substantially improve
these results.
I n the end, I like to apologize for not mentioning the work of many
2
See footnote, p. 1211.
1214
Section 1 1 : S. Klainerman
other authors. I n particular I have left out a lot of interesting results
concerning semilinear equations i.e. F = F(u) in (l') f For an up to date
bibliography concerning such results I refer to the recent papers of E. Glassey [5],, [6].
r
References
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[2] Choquet-Bruhat Y. and Christodoulou D., Existence of Global Solutions of the
Yang-Mills, Higgs Fields in 4-Dimensional Minkowski Space, I, I I , Comm. Math.
Physics 83 (1982), p p . 171-191, pp. 193-212.
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[5] Glassey R., Existence in the Large for n w — F(u) in Two Space Dimensions
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BF (ut)
[8] John F., Blow-up of Badial Solutions of Hu —
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but
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[10] John F . , Formation of Singularities in One-Dimensional Nonlinear Wave Propagation, Oomm. Pure Appi. Math. 27 (1974), pp. 377-405.
[11] John F., Improved Estimates for Blow-up for Solutions of Strictly Hyperbolie
Equations in Three Space Dimensions, in preparation.
[12] John F., Lower Bounds for the Life-Span of Solutions of Nonlinear Wave Equations in Three Space Dimensions, to appear in Comm. Pure Appi. Math., 1983.
[13] J o h n F . , Klainerman S., Almost Global Existence to Nonlinear Wave Equations in Three Space Dimensions, O.P.A.M, 1984.
[14] Kato T., Linear and Quasilinear Equations of Evolution of Hyperbolie Type,
C.I.M.E. I I CICLO, 1976.
[15] Klainerman S., Global Existence for Nonlinear Wave Equations, Oomm. Pure
Appi. Math. 33 (1980), p p . 43-101.
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Dimensions, in preparation.
[17] Klainerman S., Long-Time Behaviour of Solutions to Nonlinear Evolution Equations, Arch. Bat. Mech. and Anal. 78 (1982), pp. 73-98.
[18] Klainerman S. On Almost Global Existence to Nonlinear Wave Equations in Three
Space Dimensions, to appear in Oomm. Pure Appi. Math., 1983.
[19] Klainerman S., Weighted L°° and L1 Estimates for Solutions to the Equation in
Three Dimensions, to appear in Oomm. Pure Appi. Math., 1984.
Long Time Behaviour of Solutions to Nonlinear Wave Equations
1215
[20] Klainerman S. and Majda A., Formation of Singularities for Wave Equations
Including the Nonlinear Vibrating String, Oomm. Pure Appi. Math. 33 (1980),
p p . 241-263.
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Evolution Equations, Oomm. Pure Appi. Math. 36 (1983), p p . 133-141.
[22] Lax P . D., Development of Singularities of Solutions of Nonlinear Hyperbolic
Partial Differential Equations, J. Math. Phys. 5 (1964), p p . 611-613.
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Added in proof.
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